Shivam Patel

The two congruences for $DSOME(n)$ For a nonnegative integer $n$, let $DSOME(n)$ be the sum of the odd parts minus the sum of the even parts, taken over all partitions of $n$ into…

The spanning-tree formula for the Fitting ideal Let $\Bbbk$ be an algebraically closed field of characteristic zero, and let $\Gamma$ be a connected simple graph on $[n]$, where…

A proof of the character-sum identity for every partition We prove Hemmer–Straub–Westrem, Conjecture 6, numbered Conjecture 2.3 in the preprint. No restriction on the part sizes is…

A counterexample to the partition-graph de-anonymization conjecture The comparison in Infeld, Conjecture 1 fails even when every user and every behavior has degree $4$. The counter…

A strict majority bound for the maximizing index.

Full proof of the generalized Markoff solution-count conjecture Let … and suppose that $p$ is prime, $q=p^r$ with $r\geq1$, and $p\nmid a_{11}a_{22}a_{33}d$. Define … We prove all…

Two adjacent pendant stars violate the golden-ratio balance threshold Conjecture 5.10 of Bai, Discrete Einstein metrics on unicyclic graphs asserts that a fixed decoration of a lon…

Claim. For every nonempty finite subtraction set $S\subseteq\mathbb N$, every heap size $h\geq 0$, every pair of deterministic tie-breaking conventions, and either player $i$, the…

A uniform twelve-point counterexample to norm monotonicity Conjecture 3.6 of Liu and Slade, arXiv:2605.15545v1, is false. Take the step distribution uniform on … and take $x=(4,3)$…

Source. Gaurav Bhatnagar and Amritanshu Prasad, A bibasic double sum extension of a q-binomial theorem arising out of subspace enumeration, Conjecture 5, equation (2.3). The distin…

Source. Persi Diaconis, Andrew Lin, and Arun Ram, Schur–Weyl duality for diagonalizing a Markov chain on the hypercube, Conjecture 6.3. We also use and explicitly credit the coordi…

A 5-regular counterexample to the curvature-dimension self-improvement conjecture Conjecture 2 of Blachar, Pajot and Salez, Edge-regular graphs with non-negative curvature have pol…

Theorem. For every integer $n\geq1$,

Precise source. In Axial Morphology of the Partition Graph: Self-Conjugate Axis, Spine, and Concentration, Fedor B. Lyudogovskiy conjectures that, for each of the three invariants

Source. Rashid Barket, Enrico Grimaldi, Yacoub Hendi, Edward Hirst, Adam Onus, and Harmeet Singh, Learning the Graphical Nature of Symmetries, arXiv:2607.12026v1, Section 4.4, Conj…

Source and version. Sarah Klanderman, Katy McDicken, and Amelia Tebbe, Conditions for Building Generalized Action Graphs From Sequences, The PUMP Journal of Undergraduate Research…

Source and essential prior attribution. In Conjecture 1.1(i) of Zhi-Wei Sun, A family of polynomials and related congruences and series, Frontiers in Combinatorics and Number Theor…

Problem. MathDB 373536, the hook-length bias conjecture for 5-core partitions.

Problem. MathDB 374083, “The supertoken graph independence-number conjecture for bipartite graphs.”

Problem. MathDB 367436.

Problem: MathDB 358167, Conjecture 1 in Assaf Hallak and Gal Dalal, On the Products of Stochastic and Diagonal Matrices, arXiv:2304.11634, April 2023.

Problem: MathDB 369086, Aurora Hiveley, Experimenting with Permutation Wordle, Journal of Integer Sequences 29 (2026), Article 26.1.7, Conjecture 6; originally Conjecture 4.2 in ar…

Problem. MathDB 376190, the Low–Roberts conjecture on connected simple Hartke magic graphs.

Problem: MathDB 372912.

Problem. MathDB 382424, “Functional equation conjecture for well-aligned pairs.”